4.3 Examples
4.3.2 Geometrically Realized Models
As another controllable model, we study geometrically induced supersymmetry breaking configura- tion in Type IIB string theory onA2-fibered geometry. This has been studied in [87] somewhat in
a different context. ConsiderA2 fibred geometry [66] defined by
x2+y2+z(z−m1(t−a1)) (z+m2(t−a2)) = 0.
There are three singular points,t=a1,2anda3= (m1a1+m2a2)/(m1+m2). WrappingNcanti-D5 and Nf D5 branes on two S2s that resolve the singularities at t=a1 anda2 respectively, we can construct a supersymmetry breaking configuration. We do not wrap any brane att =a3, because this can decay into a lower energy configuration. Our present setup does not have an unstable mode as has been discussed in [90]. Therefore we can take field theory limit. Here we claim that as long as the size of S2 at t = a2 is much bigger than that at t =a1, there is a field theory description for this brane/anti-brane system. According to the conjecture proposed in [43, 82, 87, 91], there is a glueball description with respect to the hidden sector gauge group corresponding to a partial geometric transition. Thus it is reasonable to claim that the low energy field theory description is an interacting productU(1)×U(Nf) gauge theory. The kinetic term is
Z
d4θK(S1)Φ2Φ†2+ Im Z
d4θS¯1∂F0,0(S1)
∂S1
+
Z
d2θ[τ(S1)WαWα] +c.c.+. . . , (4.62)
where. . .includes higher derivative terms andU(1) gauge kinetic terms. F0,0(S1) is the prepotential for the geometry after the transition,
2πiF0,0= S12 2
log
S1
m1Λ20
−3 2
.
where Λ0is a cutoff scale of this description. The superpotential terms are
W =αS1+Nc
∂F0,0
∂S1
+ ˜W2(Φ2, S1). (4.63)
In application to phenomenology we will identify a subgroup of SU(Nf) as the standard model gauge group. So the adjoint field Φ2 forSU(Nf) gauge group should be integrated out by taking m2→ ∞. In the limit, the superpotential ˜W2 becomes a relatively simple function,
W˜2= Z a2
Λ0
"
m1
2 (t−a1) +1 2
r
m21(t−a1)2−4S1 m1
# dt.
Our goal in this section is to compute the function τ(S1) and extractCis andB1/2 from it. In the open string description we can say that thisS1 dependence is generated by the bifundamental matter. On the other hand, after the transition in closed string point of view, it is generated by closed string modes. To extract the interacting part, we use the glueball description for U(Nf) gauge group as well and assume that the glueball fields andU(1)⊂U(Nf) gauge supermultipletwα
are background fields. Following [92, 93], we use glueball description for evaluating the interacting part even though the SU(Nf) theory is weakly coupled and is not confined. Turning on these backgrounds modifies the geometry slightly. At the leading order of the modification, we read off the kinetic term for the gauge group. The low energy description is given by
L = Im Z
d4θS¯i∂F0
∂Si
+ Z
d2θ1 2
∂F0
∂Si∂Sj
wiwj
+ Z
d2θW(Si) +c.c. , (4.64)
whereW(Si) is Gukov-Vafa-Witten superpotential [81] generated by the flux. Solving the equation of motion forF1, we obtain the potential
V = 1 g11
g12F¯2+∂1W
2−g22F2F¯2−F2∂2W−F¯2∂¯2W ,¯ (4.65)
where we ignoredU(1) gauge fields. The metric is defined by Im∂i∂jF0. Since we are interested in coefficients of correlation functionsB1/2 andCi, which are related to linear terms inS2 andF2, we can put these to be zero when we evaluate the minimum of the potential,
V(S2= 0, F2= 0) = 1 g11
∂1W
2.
To find the minimum it is useful to expand the prepotentialF0forA2geometry as
F0=
∞
X
b=0
S2bF0,b(S1),
where we ignoredS1 independent part, which can be combined with the classical action ofSU(Nf) and construct the one-loop running coupling constant. In the matrix model computation,F0,b are contributions of diagrams with b boundaries, which are perturbatively calculable order by order.
With this expansion, the superpotential and metric become
W(S2= 0) = α1S1+Nc
∂F0,0(S1)
∂S1
+NfF0,1(S1), g11(S2= 0) = Im∂2F0,0
∂S1∂S1
.
In our setup, the disk and annulus amplitudes are exactly known [94–96],
2πiF0,1=S1
log
∆ +q
∆2−4Sm1
1
2Λ0
+ ∆
∆ +q
∆2−4Sm1
1
−1 2
'S1log ∆ Λ0
− S12
2m1∆2 +. . . , 2πiF0,2=1
2log
∆ + r
∆2−4S1
m1
−1 2log
2
r
∆2−4S1
m1
' S1
2m1∆2 +. . . ,
where ∆ =a1−a2. With these expressions, we see thatW(S2= 0) reproduces the superpotential in (4.63) in the limitm2 → ∞andS2 →0. At the leading order, the minimum of the potential is given by
hS1i|Nc|= (m1Λ0)|Nc| ∆¯
Λ¯0 Nf
e2πi¯α1. (4.66)
Note thatNf >0> Nc. Since there is an exponential suppression factor, vev ofS1exists in physical region, which we regard as a dynamical scale of the theory on the anti-D5 branes.
Expanding the potential (4.65) around the minimum we can read off coefficients of linear terms inS2which yield the gaugino mass term forSU(Nf) part,
2πi
g2Y Mmλ=2πiF1
16π2
−|Nc| ∂2F0,1
∂S1∂S1
+ 2Nf
∂F0,2
∂S1
hS
1i
− |F1|2 32π2iΛ4
∂2F0,1
∂S1∂S1
hS
1i
' 1 16π2
|Nc|+Nf
m1∆2 F1+ |F1|2 2i m1∆2Λ4
, (4.67)
where we supplied a dimensionful parameter Λ. In the field theory limit, we take the string scale to be infinity, keeping the scale Λ finite which should be identified with the scale ofS1 in (4.66) in our model. Theα1, which is the size of P1, also has to scale appropriately [78].2 The vev ofF1 is cut-off independent and a finite quantity in the limit,
F1=−g−111∂1W
0'βΛ4,
where the β is defined by 2iIm ¯α1 ∼ βlog Λ0,3 which encodes geometric data of the P1. On the
2In the geometric engineering one focuses on the leading effect of the small parameter Λ/Mst. Geometric quantities scale with the small parameter, for example the potential scalesV ∼(Λ/Mst)4. Thus the original string scale in the potential cancels and it becomes field theory scale vacuum energyV ∼ O(Λ4).
3Note that without loss of generality we can take the phase of ∆/Λ0 to be real. With this normalization, we defined theβ.
other hand, another correlation functions can be read off from the linear term inF2.
2πiCi(0) = −2πi 16π2Re
Im
∂F0,1
∂S1
Λ−4F¯1+|Nc|∂F0,1
∂S1
−2NfF0,2
hS
1i
' 1 16π2
Im
hS1i m1∆2
Λ−4ReF1+ (|Nc|+Nf)Re hS1i
m1∆2
. (4.68) Finally let us comment on the diagrammatical computation of the gaugino mass. Although our present geometric configuration does not include an unstable mode, we do not know explicitly the UV Lagrangian for the brane/anti-brane system. Thus it is not easy to compute the correlation functions studied above from matrix model computations directly. However, the flop of the S2 wrapping the anti-brane is a smooth process because its physical volume can never be zero [87, 91].
The new geometry yields the brane/brane configuration. The world volume theory on the branes is quiver gauge theory with a superpotential,
WSU SY = m1
2 tr(Φ1−a1)2+m2
2 tr(Φ2−a2)2+Q12Φ2Q21+Q21Φ1Q12.
Using this explicit Lagrangian and technology developed in [92, 93, 97, 98], we can compute the non- perturbative effect from perturbative Feynman diagram computations. In fact, explicit formulae for F0,0,F0,1 andF0,2 have been perturbatively computed by this method.
Part II
Gauge/Gravity Dualities
and Their Applications
Chapter 5
Introduction
We have seen in Part I of the thesis that supersymmetric gauge theories play a vital role in con- structing a realistic model for a theory above the TeV scale. But there are many known cases where supersymmetric gauge theories are related to gravity theories. In the second half of the thesis, we will discuss the understanding and applications of gauge/gravity dualities. The most well-studied example is the correspondence between the AdS5×S5 supergravity and N = 4 super Yang-Mill theory [8–10]. The correspondence can be thought of as an equivalence between the two descriptions describing the low energy dynamics ofN multiple parallel D3-branes in flat space in type IIB string theory. The low energy limit can be equivalently thought of as keeping the energy scale fixed while sending the string lengthls=√
α0to 0. On the one hand, the system is described by open strings on the D3-branes and the closed strings in the ten-dimensional bulk. In the low energy description, the D3-branes is described byN = 4SU(N) supersymmetric Yang-Mills theory and it decouples from the free bulk dynamics described by closed strings. The gauge coupling constant becomesg2Y M =gs
wheregsis the string coupling constant. On the other hand, we may view the D3-branes as a source of the energy-momentum tensor and consider its effect on the metric and other fields in supergravity.
The metric is then given by
ds2=f(r)−12dxµdxµ+f(r)12(dr2+r2dΩ25), f(r) = 1 + R4
r4 , R4= 4πgsα02N ,
(5.1)
wheredxµdxµis the four-dimensional Minkowski metric anddΩ25is the metric for the unit five sphere.
Sending α0 to 0 and keeping the energy fixed means keeping U = Rr2 fixed whereR4= 4πgsN α02. In that limit, the metric becomes
ds2=R2 dU2
U2 +U2dxµdxµ
+R2dΩ25. (5.2)
That is, the near horizon region of the geometry becomesAdS5×S5. In the limit, the near horizon dynamics decouples from the free bulk physics. Combining the two, it is natural to identifyN = 4 SU(N) supersymmetric Yang-Mills theory and the supergravity theory in AdS5×S5. Of course, this argument is not rigorous since we do not treat string theory non-perturbatively. Moreover, the supergravity description is valid whenRlsorgsN 1, while theN = 4 super-Yang-Mills theory is perturbatively described when g2Y MN =gsN 1 in the large N limit. Despite the difficulty, there is overwhelming evidence that the correspondence is correct: for example, operators with some amount of supersymmetry does not receive quantum corrections when the coupling constant gschanges, so it is possible to compare these operators in two different descriptions [10, 99].
Moreover it is believed that the essence of the correspondence does not depend on supersymmetry [100], so it makes sense to discuss non-supersymmetric versions of the correspondence also. It is a strong/weak duality, which makes it difficult to prove while beneficial to use. For example, we can learn about a strong-coupling behavior of a field theory by studying classical solutions of its gravity dual. Also, a gravity theory is lacking a UV definition, and the dual field theory may provide a way to define the gravity theory rigorously.
If we assume gauge/gravity dualities, we can study the strong-coupling behavior of some field theories by its classical dual gravity solutions. Even though we do not know an exact pair, sometimes a classical gravity geometry is determined by symmetries of the corresponding field theory to a great degree. Hence we can extract much information for a field theory if we assume the existence of a gravity dual.
Along this line, we will consider a field theory with Schr¨odinger symmetry, which is a nonrelativis- tic version of scale symmetry [101]. The Schr¨odinger symmetry is an extension of the nonrelativistic Galilean symmetry [102, 103]. Just as in the relativistic scale symmetry, there is a dilatation opera- tor, by which time and space scale differently. However, unlike its relativistic counterpart, there is only one special conformal operator. One of the most interesting physical examples with such sym- metry is a set of fermions in an optical lattice with the magnetic field. The magnetic field induces Feshbach resonances and the strength of attraction is tunable arbitrarily [104–107]. The interaction of the fermion gas arises mainly through the binary s-wave collisions with the scattering length a.
When the attraction between the fermions is very weak, the fermions favor to form Cooper pairs, forming a Bardeen-Cooper-Schrieffer (BCS) state. On the other hand, if the attraction is sufficiently strong, two fermions form a bound state and the system is effectively described by the Bose-Einstein condensation (BEC). In both the BCS and BEC limits, the system can be described as a weakly interacting system with the interaction parameter akF, where kF is the Fermi momentum. As we change the magnetic field strength, there is an intermediate regime, called the unitarity limit, where the scattering lengthabecomes infinite. In this regime, we expect to see a nonrelativistic version of scale symmetry, i.e., Schr¨odinger symmetry. Perturbation theory does not work well here sinceakF
diverge, but the gauge/gravity correspondence may provide a useful technique to study the problem.
Another application of gauge/gravity dualities is to study phase transitions of field theories.
Finite temperature states in a field theory correspond to black hole solutions in the dual gravity theory [108]. A non-zero charge density solution in the field theory can be realized by turning on the chemical potential in the grand canonical ensemble. This corresponds to a charged black hole solution in the gravity theory. An instability of a black hole signals a phase transition in the corresponding field theory. The instability may occur due to charged or neutral scalar fields as discussed in [109–111]. But a supergravity theory typically has Chern-Simons terms, and it may cause an instability [112], which we will verify in Chapter 7.
Let us consider a five-dimensional gravitational system with the Maxwell field and its Chern- Simons term. By the gauge/gravity duality, the system is dual to a four-dimensional gauge theory.
The effect of the Chern-Simons term can be analyzed as follows [10]. Suppose the geometry of the gravity solution is of the formM×X whereM is a five-dimensional space that asymptotes toAdS5
and X is some compact five-dimensional space. The Chern-Simons term appears in the gravity action as
SCS=k Z
M
A∧dA∧dA , (5.3)
where k is some constant. Since we are going to study the Maxwell field, we confine ourselves to the consideration of abelian gauge group only, even though the extension to the non-abelian gauge group is straightforward. Note that the spaceM, being asymptoticallyAdS5, has a boundary∂M, which means that the action (5.3) is not gauge invariant: under the gauge variationδA=dΛ for a zero-form Λ,
δSCS =k Z
∂M
Λ∧dA∧dA . (5.4)
Note that the changeδSCSof the Chern-Simons action depends only on the values of the gauge field on the boundary. From the view point of the boundary field theory, there is aU(1) global symmetry corresponding to the U(1) gauge symmetry of the gauge field A in the bulk. The corresponding global currentJ is coupled to an external source fieldAon the boundary. The change of the action (5.4) can be thought of as the change of the action of the boundary theory. The global currentJ couples to the external source fieldAin the formR
AµJµd4x. Under the change of the source field δAµ =∂µΛ, this term changes by −R
Λ∂µJµd4xafter partial integration. Therefore, we obtain the relation ∂µJµ ∼ kF ∧F, which tells us that the effect of the chiral anomaly of the global U(1) symmetry is proportional to the coefficient of the Chern-Simons actionk.
Given a gravity action with Chern-Simons term, we may consider a charged black hole solution.
There is the Reissner-Nordstr¨om black hole in a gravity theory without Chern-Simons term. The additional terms in the equations of motion due to Chern-Simons term vanish in that background.
Therefore, we may think that the Reissner-Nordstr¨om black hole is still a valid solution in the
presence of Chern-Simons terms. However, we will show later that the fluctuation analysis shows that there are unstable modes, depending on the Chern-Simons coupling and temperature. Such an instability is interesting since it exhibits a spatially modulated phase. In condensed matter physics, a spatially modulated phase, called the Fulde-Ferrell-Larkin-Ovchinnikov phase, occurs when two kinds of fermions with different Fermi surfaces condense with non-vanishing total momentum [113, 114].
[115] studied a similar effect in QCD. Also, in finite density QCD, the chiral density wave studied in [116, 117] exhibits such a spatially modulated phase. In addition, the Brazovskii model [118]
generates a spatially modulated phase, and it has been applied to a variety of physical problems [119].
In this model, a non-standard dispersion relation is postulated so that the fluctuation spectrum has a minimum at non-zero momentum. Gravity theories with the Chern-Simons term may provide dual descriptions for these systems.
The organization of part II is as follows. In chapter 6, we construct M-theory supergravity solutions with the nonrelativistic Schr¨odinger symmetry starting from the warpedAdS5metric with N = 1 supersymmetry. We impose that the lightlike direction is compact by making it a non-trivial U(1) bundle over the compact space. In chapter 7, we show that, in a gravity theory with a Chern- Simons coupling, the Reissner-Nordstr¨om black hole in anti-de Sitter space is unstable depending on the value of the Chern-Simons coupling. The analysis suggests that the final configuration is likely to be a spatially modulated phase.
Chapter 6
Supersymmetric Nonrelativistic Geometries in M-theory
In this chapter, we are going to consider an example of gauge/gravity dualities applied to the study of a strongly coupled field theory. We construct M-theory supergravity solutions with Schr¨odinger symmetry starting from the warpedAdS5 metric withN = 1 supersymmetry.
We first recall what Schr¨odiner symmetry is [120]. Let us start with the Galilean algebra in (1+d) dimensions consisting of the particle number operatorN, the HamiltonianH, spatial momentaPi, rotationsMij and Galilean boostsKi. The last symmetry acts on the spacetime as
t→t , xi→xi−vit , for some constant vector vi. (6.1)
An interesting feature of this algebra is that it is a subalgebra of the Poincar´e algebra in 1 + (d+ 1) dimensions. That can be most easily shown by introducing the light-cone coordinatesx±=x0±xd+1. Then we consider a subalgebra of the Poincar´e algebra that commutes with the lightcone momentum P˜−, where tilde denotes elements in the Poincar´e algebra. Then the following identification can be made:
N =−P˜−, H =−P˜+, Pi= ˜Pi, Mij = ˜Mij , Ki= ˜M−i. (6.2) Just as the Poincar´e algebra can be extended to include scale symmetry generator ˜D, we can add a dilatation generatorDto the Galilean algebra by the identificationD= ˜D+ 2(z−1) ˜M−+ for some number z, called the dynamical exponent. The commutation relation of D with other generators are
[D, Pi] =−iPi, [D, H] =−izH , [D, Ki] =i(z−1), [D, N] =i(z−2)N .
(6.3)
Note that, whenz= 2, the particle number operatorN commutes with all other generators. In that case, we can extend the algebra further by adding a special conformal transformation generatorC
that satisfies
[C, Pi] =iKi, [D, C] = 2iC , [H, C] =iD . (6.4) The special conformal generatorC can be identified with −K˜− in the Poincar´e algebra. The final algebra that containsD andC in addition to the Galilean algebra is the Schr¨odinger algebra.
Note that the Schr¨odinger algebra can be thought of as a subalgebra of the Poincar´e algebra in one higher dimension, such that its elements commute with the lightcone momentum ˜P−. P˜− is identified with the particle number generatorN, which takes discrete values. Therefore, it is natural that the direction associated with ˜P− in the gravity dual is compact. We impose this condition by making it a non-trivialU(1) bundle over the compact space.
One motivation to consider M-theory supergravity to find a solution with Schr¨odinger symmetry with non-trivialU(1) bundle is that the mass-deformed limit of three-dimensionalN = 8 maximally supersymmetric gauge theory has a concrete description in M-theory supergravity [121, 122]. Let us first see what developments have been made in the understanding of the mass-deformed theory in the field theoretic Lagrangian description.
The Lagrangian description of the maximally supersymmetric gauge theory in three dimensions was found by Bagger and Lambert [123–125] (see also [126]), by developing the idea of [127]. However, it was difficult to increase the rank of the gauge group. This is in some sense related to the fact that the maximally supersymmetric M2-brane solution does not have an adjustable parameter. Later, Aharony et al. [128] constructedN = 6U(N)×U(N) Chern-Simons-matter theory (ABJM theory) that describes multiple M2-branes on the orbifoldC4/Zk, wherekbecomes the level of the Chern- Simons action in the field theory. This orbifold provides us with an adjustable parameter, which enables us to treat weakly coupled field theories in some limit. A mass-deformed version of ABJM theory was considered in [129] and its vacuum structure was identified in [130]. Especially, in the most symmetric vacuum, the system has SU(2)×SU(2)×U(1)×Z2 symmetry. The mass term breaks the relativistic scaling symmetry. However, there is a nonrelativistic limit of this theory that has the Schr¨odinger symmetry [131, 132].
Turning our attention to the gravity side, we can turn on an anti-self-dual four-form flux for multiple M2-branes in flat space, which corresponds to adding a fermionic mass term to the field theory. The four-form flux polarizes M2-branes into M5-branes [121, 133] and the discrete set of vacua of the theory has a one-to-one correspondence with the partition of N, the number of M2- branes [122]. For multiple M2-branes on the orbifoldC4/Zk, we do not have a clear answer yet, but expect that a similar kind of solutions with desirable properties may be found.
Note that the Chern-Simons-matter theory is a good model to study the nonrelativistic limit since gauge fields are not propagating. Therefore, it is natural to seek for a supergravity solution that corresponds to the nonrelativistic limit of the mass-deformed ABJM theory. Assuming the clas-